EFFECTIVE DESIGN AND OPERATION OF

SUPPLY CHAINS FOR REMNANT INVENTORY

SYSTEMS

by

Zhouyan Wang

BS, Shanghai Jiaotong University, Shanghai, P. R. China, 1997

MBA, Shanghai Jiaotong University, Shanghai, P. R. China, 2001

MS, University of Pittsburgh, Pittsburgh, PA, 2003

Submitted to the Graduate Faculty of

the School of Engineering in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

University of Pittsburgh

2006

UNIVERSITY OF PITTSBURGH

SCHOOL OF ENGINEERING

This dissertation was presented

by

Zhouyan Wang

It was defended on

February 9th 2006

and approved by

Andrew J. Schaefer, Assistant Professor, Departmental of Industrial Engineering

Jayant Rajgopal, Associate Professor, Department of Industrial Engineering

Matthew D. Bailey, Assistant Professor, Department of Industrial Engineering

Brady Hunsaker, Assistant Professor, Department of Industrial Engineering

Prakash Mirchandani, Professor, Katz Graduate School of Business

Dissertation Advisors: Andrew J. Schaefer, Assistant Professor, Departmental of Industrial

Engineering,

Jayant Rajgopal, Associate Professor, Department of Industrial Engineering

ii

EFFECTIVE DESIGN AND OPERATION OF SUPPLY CHAINS FOR

REMNANT INVENTORY SYSTEMS

Zhouyan Wang, PhD

University of Pittsburgh, 2006

This research considers a stochastic supply chain problem that (a) has applications in a

number of continuous production industries, and (b) integrates elements of several classical

operations research problems, including the cutting stock problem, inventory management,

facility location, and distribution. The research also uses techniques such as stochastic

programming and Benders’ decomposition. We consider an environment in which a company

has geographically dispersed distribution points where it can stock standard sizes of a product

from its plants. In the most general problem, we are given a set of candidate distribution

centers with different fixed costs at the different locations, and we may choose not to operate

facilities at one or more of these locations. We assume that the customer demand for smaller

sizes comes from other geographically distributed points on a continuing basis and this

demand is stochastic in nature and is modeled by a Poisson process. Furthermore, we

address a sustainable manufacturing environment where the trim is not considered waste,

but rather, gets recycled and thus has an inherent value associated with it. Most importantly,

the problem is not a static one where a one-time decision has to be made. Rather, decisions

are made on a continuing basis, and decisions made at one point in time have a significant

impact on those made at later points. An example of where this problem would arise is a steel

or aluminum company that produces product in rolls of standard widths. The decision maker

must decide which facilities to open, to find long-run replenishment rates for standard sizes,

and to develop long-run policies for cutting these into smaller pieces so as to satisfy customer

demand. The cutting stock, facility-location, and transportation problems reside at the heart

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of the research, and all these are integrated into the framework of a supply chain. We can see

that, (1) a decision made at some point in time affects the ability to satisfy demand at a later

point, and (2) that there might be multiple ways to satisfy demand. The situation is further

complicated by the fact that customer demand is stochastic and that this demand could

be potentially satisfied by more than one distribution center. Given this background, this

research examines broad alternatives for how the company’s supply chain should be designed

and operated in order to remain competitive with smaller and more nimble companies.

The research develops a LP formulation, a mixed-integer programming formulation, and a

stochastic programming formulation to model different aspects of the problem. We present

new solution methodologies based on Benders’ decomposition and the L-shaped method to

solve the NP-hard mixed-integer problem and the stochastic problem respectively. Results

from duality will be used to develop shadow prices for the units in stock, and these in turn will

be used to develop a policy to help make decisions on an ongoing basis. We investigate the

theoretical underpinnings of the models, develop new, sophisticated computational methods

and interesting properties of its solution, build a simulation model to compare the policies

developed with other ones commonly in use, and conduct computational studies to compare

the performance of new methods with their corresponding existing methods.

Keywords: Supply Chain, Inventory Management, Production, Distribution, Facility Loca-

tion, Integer Programming, Benders’ Decomposition, Stochastic Programming, L-shaped

Method.

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TABLE OF CONTENTS

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.0 BASIC MODEL: PRODUCTION AND DISTRIBUTION . . . . . . . . 8

2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 A Price-Directed Strategy for Dynamic Supply Chain Management . . . . 14

2.3.1 Strategy in a Dynamic Demand Environment . . . . . . . . . . . . 14

2.3.2 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.3 Degeneracy, Multiple Optima, and Perturbation . . . . . . . . . . . 19

2.4 Properties of the ²-Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Simulation and Inventory Control . . . . . . . . . . . . . . . . . . . . . . . 38

3.0 LOCATION MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1 Problem Formulation and Policy Development . . . . . . . . . . . . . . . . 42

3.1.1 Primal Formulation of the Static Problem . . . . . . . . . . . . . . 43

3.1.2 Implementation in a Dynamic Environment . . . . . . . . . . . . . 44

3.2 Benders’ Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Guaranteeing Feasibility of P-SP(ẑ) . . . . . . . . . . . . . . . . . . . . . . 50

3.3.1 Demand Size Version . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.2 Raw Size Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

v

3.4 Relationship among Various Polyhedra . . . . . . . . . . . . . . . . . . . . 58

3.4.1 Eliminating Redundancy . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.3 Relationship among Polyhedra . . . . . . . . . . . . . . . . . . . . 61

3.5 Computational Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.0 STOCHASTIC MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.1 The Single-Cut L-shaped Method . . . . . . . . . . . . . . . . . . . 72

4.1.2 The Multi-Cut L-shaped Method . . . . . . . . . . . . . . . . . . . 74

4.1.3 Overview of Our Approach . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.2 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.3 Comparison of Single and Multi-Cut Methods . . . . . . . . . . . . 83

4.3 The Double Cut L-shaped Method . . . . . . . . . . . . . . . . . . . . . . 85

4.4 Feasibility Cuts to Guarantee the Feasibility of Program SSP . . . . . . . . 87

4.4.1 New Feasibility Constraints . . . . . . . . . . . . . . . . . . . . . . 89

4.4.2 Feasibility after Using New Feasibility Constraints . . . . . . . . . . 93

4.5 Alternative L-shaped Methods . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.6 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.0 CONCLUSIONS AND FUTURE RESEARCH . . . . . . . . . . . . . . . 102

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2 Extensions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 103

APPENDIX A. COMPUTATIONAL RESULTS FOR LP MODEL . . . . . 105

APPENDIX B. COMPUTATIONAL RESULTS FOR LOCATION MODEL 107

APPENDIX C. COMPUTATIONS FOR STOCHASTIC MODEL . . . . . 111

APPENDIX D. NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . 115

D.1 Notation for Deterministic Problem . . . . . . . . . . . . . . . . . . . . . . 115

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D.1.1 Greek Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

D.1.2 Roman Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

D.1.3 Abbreviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

D.2 New Notation for Stochastic Problem . . . . . . . . . . . . . . . . . . . . . 116

D.2.1 Greek Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

D.2.2 Roman Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

D.2.3 Abbreviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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LIST OF TABLES

1 Facility Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Unit Shipping Costs and Demands . . . . . . . . . . . . . . . . . . . . . . . 18

3 Optimal Dual Prices (ηik) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Test Instance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Performance of the Policies* . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Performance of Bounds Given by Proposition 11 . . . . . . . . . . . . . . . . 41

7 Parameters of Raw and Demand Sizes . . . . . . . . . . . . . . . . . . . . . . 52

8 Facility Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9 Parameters of Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10 Performance of Direct and Benders’ Approaches . . . . . . . . . . . . . . . . 64

11 Characteristics of Tested Instances . . . . . . . . . . . . . . . . . . . . . . . . 65

12 Comparison of Three Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 66

13 Comparison between Benders and Algorithm 4 . . . . . . . . . . . . . . . . . 68

14 Performance of Single and Multi-cut L-shaped Methods . . . . . . . . . . . . 84

15 Facility Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

16 Unit Shipping Costs under Scenario One . . . . . . . . . . . . . . . . . . . . . 91

17 Unit Shipping Costs under Scenario Two . . . . . . . . . . . . . . . . . . . . 92

18 Characteristics of Tested Stochastic Instances . . . . . . . . . . . . . . . . . . 98

19 Performance Comparison for Stochastic Algorithms . . . . . . . . . . . . . . . 99

20 Performance of Three Policies and Bound Gap between Proposition 11 and

Optimal Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

21 Performance of Three Policies and Bound Gap, Cont. . . . . . . . . . . . . . 106

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22 Performance on Small Location Problems . . . . . . . . . . . . . . . . . . . . 108

23 Performance on Medium Location Problems . . . . . . . . . . . . . . . . . . . 109

24 Performance on Large Location Problems . . . . . . . . . . . . . . . . . . . . 110

25 Performance of Algorithms for Small Stochastic Problems . . . . . . . . . . . 112

26 Performance of Algorithms for Medium Stochastic Problems . . . . . . . . . . 113

27 Performance of Algorithms for Large Stochastic Problems . . . . . . . . . . . 114

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LIST OF FIGURES

1 Distribution Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Sub-Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Alternative Optima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Optimal Cutting Scheme for Original Problem . . . . . . . . . . . . . . . . . 26

5 Optimal Cutting Scheme for Perturbed Problem . . . . . . . . . . . . . . . . 26

6 Alternative Optimum for Perturbed Problem . . . . . . . . . . . . . . . . . . 27

7 Redundancy of Benders’ Feasibility Cuts . . . . . . . . . . . . . . . . . . . . 67

8 Comparison of Total Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

x

PREFACE

I first would like to appreciate the support from the National Science Foundation via Grant

DMI 0217190.

I am especially indebted to my dissertation chairs, Dr. Andrew J. Schaefer and Dr.

Jayant Rajgopal, for their continuous guidance, support, and inputs throughout my Ph.D

program. I could not have written this dissertation without their strong encouragement and

determination. Their solid academic backgrounds, invaluable comments, and kind personal-

ities have been my life long assets and were essentially helpful in shaping the foundation of

my dissertation.

I would like to express my sincere thanks to my committee members, Dr. Matthew

D. Bailey and Dr. Brady Hunsaker from the Industrial Engineering department, and Dr.

Prakash Mirchandani from the Operations, Decision Sciences, and Artificial Intelligence

department. They contributed their valuable time and efforts to review earlier drafts of my

dissertation and answered numerous questions I had. Their comments were really prompt

and insightful.

My special thanks extend to all Industrial Engineering faculty, especially Dr. Bopaya

Bidanda, Dr. Mainak Mazumdar, and Dr. MingEn Wang for teaching me knowledge and

skills to do research and to teach.

I wish to thank all lovely Ph.D students, especially Nan Kong, Jennifer Kreke, Steven

Shechter, and Lizhi Wang for many inspirational discussions.

My dissertation is dedicated to my dearest wife, Huajing Chen. Without her solid sup-

port, strong encouragement, and patient love, it would have been impossible for me to

complete my degree successfully. My wonderful and most understanding parents, my father

Shuling Wang and my mother Xueying Chen, receive my deepest appreciation for their gen-

xi

erous, unconditional, and endless love towards me. My elder brother, Zhoulu Wang, has

been a great supporter of my work and has accompanied me going through many tears and

laughters during my studies.

xii

1.0 INTRODUCTION

1.1 MOTIVATION

We consider a supply chain problem that integrates elements of several classical optimization

problems, including the cutting stock problem, facility-location, inventory management, and

distribution. Relatively large pieces, called raws, are manufactured or bought from suppliers

and stocked at distribution centers. Demand from geographically dispersed customers arises

for a variety of different lengths that must be cut at these distribution centers from the

standard-sized rolls and then shipped to customers. The overall objective is to find which

facilities to operate, long-run rates of replenishment for the standard sizes at each facility, and

long-run policies for cutting these raws and remnants into smaller pieces to satisfy stochastic

customer demands.

A large multinational steel company presented the problem addressed here. Steel is

typically produced as rolls having one of several standard lengths. These rolls are then

sent out to distribution centers that might either be attached to plants or at independent

locations. Demands arise for a variety of different lengths that must be cut from the standard-

sized rolls and shipped to customers on a daily basis. These demands are stochastic in nature

and are modeled by a Poisson process. The decision of which rolls to cut is complicated by the

fact that the distribution centers also maintain inventories of remnants generated from prior

cutting operations, unless of course, these remnants are considered too small to satisfy any

potential demand, in which case they are sold as scrap in the open market. Consequently,

it is clear that a decision made at some point in time affects both the ability to satisfy

demand at a later point in time, as well as the associated cost when the decision can be

made in more than one way. The situation is further complicated by the fact that customer

1

demand is stochastic and that this demand could be potentially satisfied by more than one

distribution center. Given this environment, the company was interested in looking at broad

alternatives for how its supply chain should be redesigned and operated in order to remain

competitive with smaller and more flexible companies.

1.2 LITERATURE REVIEW

At the heart of this research is the production-distribution and facility-location problem for

remnant inventory systems. The cutting stock problem is familiar to most researchers in

Operations Research, and hundreds of papers (e.g., [21, 36, 37, 49, 50]) have been written on

the subject and its variants since the seminal work of Gilmore and Gomory [41, 42, 43]. The

interested reader is referred to Cheng et al. [23], or Haessler and Sweeney [46] as starting

points on the aforementioned topic. The basic cutting stock problem is typically a static

one [53, 68, 69, 90]. Both optimal as well as heuristic methods have been applied to solve

the problem [97, 103, 104, 105, 110]: the product is produced in (a relatively small number

of) standard sizes, and one must meet a given set of requirements for this product. These

requirements encompass (a relatively large number of) non-standard sizes that must be cut

from the standard stock. The problem is to find the number of pieces of stock of each

standard size to use, along with the associated cutting patterns, so that all requirements are

satisfied at minimum cost [23]. This situation is very common in a number of continuous

production applications including metals such as steel or aluminum, paper, textiles, lumber,

fiber-optic and electrical cables, and glass.

The cutting stock problem presented in this research is a little different. First, we

address the problem in the context of a supply chain where a company has geographically

dispersed distribution centers that can be stocked with standard sizes that are procured from

its plants; the demand at these centers for smaller sizes comes from other geographically

distributed points. Second, unlike classical cutting stock problems, in our model decisions

are made in a stochastic and dynamic environment. Third, we account for a sustainable

manufacturing environment where the trim is not necessarily treated as waste; rather, it

2

may get recycled. This assumption holds in industries such as steel, or aluminum. Fourth,

and most importantly, the problem is not static, where a one-time decision has to be made.

Rather, decisions are made on an ongoing basis since demand is dynamic, and decisions

made at one point in time have a significant impact on those made later.

Steel can either be produced in integrated mills from iron ore, or in so-called mini-mills

from recycled scrap. Much of the relevant research has focused on optimizing steel production

planning and scheduling [78, 88, 99, 115]. A survey of this topic may be found in Tang et al.

[98], where numerous approaches for solving scheduling and planning problem in the steel

industry are reviewed. A few papers have looked at the combined production and distribution

system, e.g., Chen and Wang [22] present a deterministic linear programming model for this

integrated production and transportation planning problem. Inventory planning in the steel

industry is addressed by Denton et al. [35], who present a model of a hybrid make-to-stock,

make-to-order system to help an integrated steel mill manage its inventory. Krichagina et al.

[61] use a combination of linear programming and Brownian analysis to address a problem

in a dynamic and stochastic environment, but this paper looks at a single location, and only

considers the production problem. Denton and Gupta [34] describe a model where they use a

two-stage stochastic integer programming model to decide on optimal levels of semi-finished

steel inventory, for use with a delayed product differentiation approach.

In addition to steel, researchers have also addressed production and inventory issues

in other remnant inventory systems such as aluminum, copper, and paper and pulp. For

example, Hendry et al. [47] consider a two-stage production planning problem in the copper

industry and describe an IP-based heuristic method to solve the problem; Johnson et al. [57]

address a combined skiving and cutting stock problem in paper mills; Bredström et al. [18]

develop MIP models for the production and distribution problem in pulp mills and provide a

heuristic based on column generation; and Partington [82] addresses inventory issues related

to aluminum.

The classical transportation problem first described by Hitchcock in 1941 [51]. In the

area of integrated production and transportation, Hochbaum et al. [52] show that for the

production-distribution problem, it is still NP-hard even if demand is deterministic and the

production cost function is concave, separable and symmetric. While some authors [31, 56,

3

81, 91, 109] focus on strategic models that integrate design decisions such as location, plant

capacity, and transportation channels into the supply chain, our research is involved more

with both production and distribution operations. In this area, some authors [16, 17, 85, 113]

consider decisions across multiple stages, and others [10, 14, 15, 20, 95] address problems

that integrate inventory and distribution decisions.

For algorithms for this NP-hard production-distribution problem, the paper by Tuy et

al. [101] is a good starting reference since they look into a problem with two production

facilities. Tuy et al. [102] develop a polynomial time algorithm for the simple case of two

production facilities with concave production costs and linear transportation costs. Kuno and

Utsunomiya [62] present a Lagrangian-based branch-and-bound algorithm for the situation

where the production cost function is concave and separable. All of these papers only consider

deterministic demand.

The classical network location problems include the maximum covering location problem

[32, 79], the set covering location problem [9, 30, 100], the p-center problem [19, 70, 83, 87],

and the facility location problem [5, 8, 27, 66], which includes the uncapacitated version

(UFLP [4, 44]) as well as capacitated version (CFLP [7, 40, 60]). In general, both UFLP

and CFLP are NP-hard [77] and have been widely studied. Many algorithms and heuristics

have been developed in the past decades [7, 38, 55, 59, 106]. An exhaustive survey can be

found in [6], while a more recent review of algorithms, heuristics, and exact methods for the

capacitated facility location problem may be found in [96].

Some research has also been conducted on the polyhedral structure of the NP-hard facility

location problem. For the capacitated facility location problem, Leung and Magnanti [65]

look into the polyhedral structure for the case of constant, equal capacities; Aardal et al. [1]

study the case of different capacities; Deng and Simchi-Levi [33] consider the case of unsplit

demands; Cornuéjols at al. [28] compare the strength of various Lagrangian relaxations.

Interested readers are also referred to polyhedral study for the uncapacitated facility location

problem [24, 25, 29, 45].

The problem most closely related to ours is the multi-product capacitated facility location

problem (MPCFL), first introduced by Lee [63, 64]. Relatively few researchers have studied

4

this problem. Mazzola and Neebe [75] show that it is NP-hard and presented a solution

method based on Lagrangian relaxation method.

In this research, we extend the deterministic location-production-distribution problem to

the stochastic case. Stochastic programming has been well studied, e.g. Birge and Louveaux

[11], Higle and Sen [48], and Kall and Wallace [58]. For our problem in this case, the decision

is further complicated by the fact that there exist several scenarios in which some parameters,

i.e. the cutting costs, raw costs, demand rates, etc., will have different values under different

scenarios. Specifically, we consider facilities that are capacitated with the types, sizes, and

costs of raw materials processed, cutting costs, capacities, and scrap value possibly being

different under different scenarios. The types, sizes, and amounts demanded by customers

are also stochastic variables. Finally, the shipping costs from the facilities to the customers

can change over scenarios as well.

In examining prior research, the most closely related to this research is some relatively

recent work (Adelman and Nemhauser [2]; Adelman et al. [3]) applied to the fiber-optic cable

industry. In these papers, the authors consider a single supply and a single demand location

and build a linear programming model that allows standard duality results to place a “value”

on remnants of any size; this in turn leads to a cutting policy adapted to a dynamic demand

environment wherein one chooses the size of the piece to be cut based upon the reduction in

its value when it is cut to a remnant of smaller size.

Our work extends the work in Adelman and Nemhauser [2] in several regards. First,

unlike the fiber optic cable industry, scrap in the metals industry has value, and our models

account for this. For example, used beverage can scrap value could be as high as 75% of the

original value [80], and coil steel scrap value could be more than 25% of the original value

[86, 92]. Second, we do not assume prespecified proportions in which different raw stock

sizes are replenished, but rather, allow our model to find the optimal replenishment rates

for each location; correspondingly, the model is far more realistic in practice. Third, the

decision maker in our model incurs a cost for each cutting operation. The cutting cost is

nontrivial in the steel industry (Chu and Antonio [26]) and ignoring this cost is inaccurate in

real-life practice. Fourth, unlike Adelman and Nemhauser [2] who consider a single location,

we consider multiple supply and demand locations. The modified model is then embedded

5

within a location/distribution network for the general supply chain problem in order to

determine which distribution facilities should be selected for operation, what cutting policies

should be used, and how product should be distributed from distribution centers to demand

points. Fifth, we consider the fixed cost and the capacity limit for each facility. Rather

than opening all facilities, we also let the model itself choose the optimal set of facilities to

open from the potential locations. Sixth, unlike Adelman and Nemhauser [2] who consider

the deterministic problem, we make the model even more realistic by introducing stochastic

parameters.

1.3 SIGNIFICANCE

The research is significant in several regards. First, it represents one of the few efforts

at an exact solution to a supply chain problem that integrates location, production and

distribution. Second, the application domain is potentially vast since many industries, such

as paper, steel and aluminum, deal with remnant inventories similar to those addressed

here. Third, the research aims to improve solution procedures for large-scale stochastic

programming problems, which is an area that has been relatively unexplored because of the

computational difficulties associated with it.

In this dissertation, major contributions are as follows. First, to effectively manage long-

run costs in a dynamic and stochastic demand environment, we develop a policy to assist

price-directed policy makers in deciding which facility to satisfy the demand from and to pick

which remnant to cut to obtain the demanded size. Second, to make the policy applicable,

we develop a new perturbation technique to make the dual prices satisfy several desirable

properties. Third, considering the facility location problem results in an NP-hard mixed-

integer program. To solve the resulting large-scale integer program, we adapt Benders’

decomposition [74]. To improve its performance, we develop a new method to generate

a set of feasibility constraints that replace Benders’ feasibility cuts and whose number is

bounded polynomially. Fourth, by considering stochastic demand, we develop a stochastic

program and present new algorithms based on the L-shaped method to solve the resulting

6

problem. Fifth, it also shows the potential of applying new feasibility constraint in complex

solution methodologies because our results prove new feasibility constraints are robust in

complex solution methodologies such as Benders’ Decomposition and the L-shaped method.

Finally, our solution development provides insight into algorithmic and computational issues

regarding these methodologies.

This document is organized as follows. We first start in Chapter 2 with a linear program-

ming model for the actual cutting and distribution problem, given a set of locations, and

then present a strategy for dynamic demands. In Section 2.5, we address inventory control

and present a simulation model to evaluate the technique that is developed. In Chapter 3,

the LP model is then extended to the case where the location of distribution centers must

be decided by considering their different fixed costs and capacity limits. Solution algorithms

based on Benders’ decomposition are proposed for this problem. In Chapter 4, we present

an even more general model with stochastic parameters, and develop a stochastic program-

ming approach based on the L-shaped method to solve this problem. Chapter 5 presents

conclusions and directions for future research.

7

2.0 BASIC MODEL: PRODUCTION AND DISTRIBUTION

In our first model, we assume that all facilities are open and have no capacity limits. Each

facility stocks some set of standard raw sizes. Demand arises from the various customers

according to a Poisson process with different mean values. When a certain demand size

arises, an effective policy should help the decision makers choose an appropriate raw or

remnant at one of the facilities so that material, production, and transportation costs are

minimized in the long run. The objective of this model is to find such a policy.

2.1 SYSTEM DESCRIPTION

As shown in Figure 1, we consider a production-distribution system consisting of a set of

geographically distributed facility locations indexed by i ∈ I each of which stocks a productin one or more “standard” sizes. The product is replenished periodically from an outside

production facility. Demand for the product arises from some other set of geographically

distributed demand locations indexed by j ∈ J , but typically in smaller, “non-standard”sizes. The demand for a particular size k at a particular location j is assumed to follow a

Poisson distribution with some known mean rate λjk. We assume that there is no capacity

limit at each facility i and that all demand must be fulfilled. A cost δi is incurred each time

a cut is made at location i and a per-unit transportation cost cijk is incurred in shipping a

unit of size k from location i to location j. The problem is to determine a long-term policy

for cutting and distributing the product in order to satisfy demand.

In order to formulate the problem first note that remnants from cutting standard-size

stock can either be used again to produce other (smaller) non-standard sizes, or recycled

8

Figure 1: Distribution Network

as scrap. Thus, at any given time each distribution facility could be expected to have an

inventory of units of various sizes. Now consider a node in Figure 1 that represents some

distribution facility i. Each such node will be represented by its own sub-network. These sub-

networks are then connected to the various demand locations as part of a larger distribution

network; thus the overall network is one comprising of several smaller sub-networks. Each

of the sub-networks is represented using a model similar to that proposed in [2] for their

single supplier/single user problem. In this model, nodes in each sub-network represent the

various sizes that could be present at the facility at any time as standard (raw) stock, finished

product, remnants or scrap, where node m represents size m. The flow along an arc from

node m to node n (where n < m) represents the number of units of size m that are cut down

to remnants of size n (while producing that many pieces of size m−n in the process); Figure2 illustrates an example of such a sub-network.

9

Figure 2: Sub-Network

2.2 MODEL FORMULATION

To describe our model, we first introduce some notation.

• I = Set of facility locations.• J = Set of demand locations.• Ki = Set of all possible sizes, including scrap sizes (we refer to sizes that are smaller

than the smallest demanded size as scrap sizes), that could be generated at facility i (we

include the size 0 in this set); define K =⋃

i∈I Ki.

• Dj = Set of all sizes that are demanded at demand location j ∈ J ; define D =⋃

j∈J Dj.

• Si ⊆ Ki = Set of all raw stock sizes that are processed at facility location i ∈ I; defineS =

⋃i∈I Si.

• λjk = Mean demand rate at location j ∈ J for product of size k ∈ Dj.• cijk = Cost of transporting a single unit of size k ∈ Dj from location i ∈ I to location

j ∈ J .• aih = Cost of one unit of raw stock of size h ∈ Si at location i ∈ I.

10

• σi = Unit salvage value of product at facility i ∈ I; σi > 0.• δi = Cost per cut at facility i ∈ I; δi > 0.

The variables are:

• xijk = Rate of shipment of units of size k ∈ Dj from facility location i ∈ I to demandlocation j ∈ J .

• yim,n = Rate of generation at facility location i ∈ I, of units of size (m − n) that areobtained by cutting size m(≥ n) down to size n.

• rih = Rate of replenishment of raw units of size h ∈ Si at facility location i ∈ I; definerih = 0 if h ∈ (Ki\Si).

• siu = Rate at which size u is scrapped at facility location i ∈ I. Define siu = 0 if the sizeu cannot be generated at location i.

Before formulating the problem, we also define for each i ∈ I, the sub-network Gi =(Ki, Ai) similar to the example in Figure 2, where Ki is the set of nodes and Ai the set of

arcs in the graph. Specifically, we define:

Ai = {(m,n)|(m− n) ∈ D; m,n ∈ Ki,m > n}.

Note that for the sub-network in Figure 2, Ki = {0, 1, 2, 3, 5, 6, 8, 11} and the arc setAi = {(11,8),(11,6), (8,5),(8,3),(6,3),(6,1),(5,2),(5,0),(3,0)}.

We are now ready to formulate the problem as the following linear program (LP):

Minimize∑i∈I

∑

h∈Siaihrih+

∑i∈I

∑

{(m,n)∈Ai|n>0}δiy

im,n+

∑i∈I

∑j∈J

∑

k∈Djcijkxijk−

∑i∈I

∑u∈Ki

uσisiu (2.1)

subject to:

ril +

∑

m|(m,l)∈Aiyim,l

−

sil +

∑

n|(l,n)∈Aiyil,n

= 0, for all i ∈ I and l ∈ Ki, (2.2)

11

∑i∈I

xijk = λjk, for all j ∈ J and k ∈ Dj, (2.3)

∑

(m,n)∈Ai|(m−n)=kyim,n

−

(∑j∈J

xijk

)= 0, for all i ∈ I and k ∈ D, (2.4)

all x, y, r, s ≥ 0. (2.5)

It is readily seen that the objective (2.1) is to minimize cost of raw stock, cutting and

transportation, less the value of the scrap that is salvaged. Constraints (2.2) ensure flow

balance at each node of each facility sub-network, constraints (2.3) ensure that all demand is

satisfied, and constraints (2.4) ensure that total flow of each size out of each sub-network is

equal to its production in that sub-network. Notice that this primal formulation only yields

a replenishment and shipment policy for a static environment for one particular demand rate

(the expected value λjk). What one actually needs is a policy to use in a dynamic environment

where demand occurs in a stochastic fashion. Furthermore, additional difficulty arises from

the fact that in general the above LP will have multiple optima.

Figure 3: Alternative Optima

For instance, consider Figure 3: If the flow in the Figure 3(a) is optimal, then so is the

one in Figure 3(b). Both take in 25 units of raw stock of size 11 and generate 30 units of size

12

5 and 20 units of size 6, along with 5 units of scrap of size 1. While there is no difference

between one optimum and the other for the static problem formulated in this section, the

same cannot be said if the objective is to come up with a long-term operating policy for the

case where the demands are dynamic. In such an environment a particular decision could

have a significantly more (or less) adverse impact on future decisions.

In light of the preceding discussion we introduce the dual program corresponding to the

above LP. Define the dual variables:

• ηi,m corresponding to each constraint in (2.2), i.e., to each node of each facility’s sub-network,

• µjk corresponding to each constraint in (2.3), i.e., to each demanded size at each location,• πik corresponding to each constraint in (2.4), i.e., to each demanded size produced at

each location.

Then the corresponding dual is:

Maximize∑j∈J

∑

k∈Djλjkµjk, (2.6)

subject to:

µjk − πik ≤ cijk, for all i ∈ I, j ∈ J, k ∈ Dj, (2.7)

ηi,n − ηi,m + πik ≤ δi, for all i ∈ I; m,n ∈ Ki; (m, n) ∈ Ai; k = m− n, n > 0, (2.8)

ηi,n − ηi,m + πik ≤ 0, for all i ∈ I; m,n ∈ Ki; (m,n) ∈ Ai; k = m− n, n = 0, (2.9)

ηi,h ≤ aih, for all i ∈ I, h ∈ Si, (2.10)

ηi,u ≥ uσi, for all i ∈ I, u ∈ Ki, (2.11)

η, µ, π unrestricted.

In order to interpret the dual, consider an external entity that is willing to deliver the

demanded pieces directly to the customer at the demand point. Then µjk may be interpreted

13

as the sale price set by this entity for a unit of size k at demand point j. The dual objective

is to maximize revenues. The same entity is also willing to supply pieces at distribution

facility i for the price of πik for a piece of size k. We may interpret ηi,m as the inherent unit

value of a product of size m at facility i. Note that the ηi,m values provide us with the value

associated with a particular size of remnant. For our long-term decision-making, these are

the values that will be used.

Then (2.7) states that the sale price at location j can be no higher than the price

charged at facility i plus the shipping cost from facility i to demand location j (otherwise

the decision maker would buy at the facility at the price of πik and ship it himself). To

interpret constraints (2.8) and (2.9) first note that ηi,m - ηi,n is the loss in inherent value

when a piece of size m is cut down to a piece of size n (while producing a piece of size

k=(m−n) in the process). Thus the quantity ηi,m - ηi,n +δi may be interpreted as the “totalcost” to us of making a piece of this size k. Then (2.8) and (2.9) state that the price charged

at facility i for size k cannot exceed the cost to make it (otherwise the decision maker would

make it himself and not buy from the external entity); note that in (2.9) we do not include

cutting costs since the entire remnant is used and there is no cutting operation. Constraints

(2.10) and (2.11) are defined for each facility; (2.10) says that a unit of size k cannot be

worth more than its raw unit cost; and (2.11) says that a unit of size k is worth at least as

much as its scrap value.

2.3 A PRICE-DIRECTED STRATEGY FOR DYNAMIC SUPPLY CHAIN

MANAGEMENT

2.3.1 Strategy in a Dynamic Demand Environment

In the previous section we have presented primal-dual formulations for a static version of

our problem, i.e., we have assumed a fixed value for the demand rate for each size at each

location. In practice, demand is stochastic and is modeled by a Poisson process with some

rate vector λ. Thus, what one needs is a policy that can be used to make demand fulfillment

14

decisions in this dynamic demand environment. To do this, we examine the optimal solution

to the static problem and use it to arrive at a sensible policy for the dynamic problem. The

dual program just described is crucial to this step since the policy will be based on the values

of the optimal dual variables.

As an example of what this entails, suppose that a customer at demand location j orders

a unit of size 5. There may be multiple facilities from which this demand could be satisfied

as well as multiple cutting options at a particular facility. Since a decision made today will

influence the cost of a decision made tomorrow, the questions to be answered are of the

following kind: “Should we cut this order from our 11-unit inventory at Chicago, from our

13-unit inventory at Chicago, or from our 11-unit inventory at Pittsburgh?” Recall that ηi,m

is the dual variable corresponding to constraint (2.2) for a remnant of size m at location i.

Thus the optimal (static) value for ηi,m represents the marginal value of an extra piece of

size m at location i, i.e., the inherent value of this particular remnant. There is intuitive

appeal in using this value as the basis for a policy that is adapted to the dynamic demand

environment. In particular, the answer to the question posed above will depend on (a) the

reduction in inherent value when going from a larger to a smaller size (e.g., 13 to 8 units

or 11 to 6 units, as represented by ηi,13 − ηi,8 or ηi,11 − ηi,6, respectively), (b) the cuttingcosts δi at Pittsburgh and Chicago, and (c) the transportation cost from the source to the

destination.

The fundamental question to answer is the following: “What is the general policy that

we should follow for choosing from multiple cutting and distribution options in order to

minimize long-run costs?” Obtaining a dual solution to the static problem allows us to

develop such a policy for the dynamic case and also provides us with information as to when

we might be indifferent between one policy and another.

Theorem 1. Suppose there is demand at location j for units of size k (that is, k ∈ Dj andλjk > 0). The optimal static strategy for fulfilling this is to choose the location i

′ and cutting

size m′ via: (i′, m′) ∈ argmini∈I,(m∈Ki|m≥k)(ηi,m − ηi,m−k + δi + cijk), where δi = 0 if m = k.

Proof. Consider the set Ik = {i ∈ I|∃m ≥ k, yim,m−k > 0} at optimality. That is, Ik is theset of all facilities that produce units of size k according to the optimum solution. We would

15

like to see which of these facilities could possibly have xijk > 0 at optimality, i.e., be used to

fulfill the demand arising at location j for size k. Suppose then that i′ ∈ Ik is such a facility,i.e., xi′jk > 0 at optimality.

From constraint (2.7) of the dual problem we know that µjk ≤ πik + cijk for all i ∈ I.Since (a) µjk appears only in (2.7), (b) λjk ≥ 0, and (c) the dual objective is to be maximized,it also follows that

µjk = mini∈I

(πik + cijk). (2.12)

Now consider complementary slackness conditions corresponding to (2.7) and (2.8):

(µjk − πik − cijk)xijk = 0, for all i ∈ I, j ∈ J, k ∈ Dj (2.13)

(ηi,n − ηi,m − δi + πik)yim,n = 0, for all i ∈ I, (m,n) ∈ Ai, k = m− n (2.14)

Since xi′jk > 0, complementary slackness condition (2.13) implies µjk = πi′k + ci′jk, so

that from (2.12), i′ ∈ argmini∈I(πik + cijk). Also, from complementary slackness condition(2.14), for all i ∈ Ik, if size m′ is cut down to size n = (m′−k), (so that yim′,n > 0), it followsthat πik = ηi,m′ − ηi,(m′−k) + δi. Since all defined cijk > 0 it follows from the two previoussentences that (i′,m′) ∈ argmini∈I,(m∈Ki|m≥k)(ηi,m − ηi,(m−k) + δi + cijk).

Theorem 1 implies that if there is demand for size k, the optimal strategy in the static

problem is to consider all pieces of size m ≥ k at all facilities and pick a facility and size forwhich (ηi,m − ηi,m−k + δi + cijk) or (ηi,m − ηi,0 + cijk) is minimized, depending on whetherm > k or m = k respectively. The intuition behind this is straightforward: the difference

ηi,m − ηi,m−k is the decrease in value to the decision-maker associated with reducing a unitof size m to size m − k at facility location i and, along with δi, thus represents a “cost.”The additional term cijk is added to account for the transportation cost from facility i to

demand location j. Furthermore, since we assume no limits on capacity, all of the demand

from customer j for size k would be satisfied from the same facility i.

We use Theorem 1 as the basis for our policy in a dynamic demand environment by re-

stricting our search to the existing inventory at each facility. The optimal size corresponding

16

Table 1: Facility Data

Facility 1 Facility 2

Raw Unit Cutting Raw Unit Cutting

Size Cost Scrap Value Cost Size Cost Scrap Value Cost

13 20 0.05 0.05 11 19 0.1 0.05

11 19

to a static policy might not be in stock at the associated facility and there is thus no guar-

antee that the optimal static policy will always be used in a dynamic demand environment.

However, it does provide us with a logical basis for a policy in the dynamic environment.

Such policies based on the inherent values of inventory are referred to as price-directed

policies. We illustrate this policy by a small numerical example in the next section.

2.3.2 A Numerical Example

In this section, we provide a small numerical example to show how the policy works. Consider

a system with two facilities and two demand locations. The raw sizes available at each

facility along with the unit costs are summarized in Table 1. This table also provides the

unit salvage values at each facility. The demanded sizes and the corresponding demand rates

at each demand point, along with unit shipping costs are summarized in Table 2.

Upon formulating and solving this problem we obtain the dual prices given in Table 3.

Note that there are certain sizes that can never be produced as remnants, and the ηi,k are

not defined for these.

To see how the solution translates into a dynamic policy, consider a case where there is

demand for 3-unit pieces at Location 2. Suppose that at the time that this demand arrives,

we have 13-unit pieces and 6-unit remnants at Facility 1 and 6-unit remnants at Facility 2. In

this case, facility 1 has: η1,13−η1,10+c123+δ1 = 20−17+2+0.05 = 5.05, η1,6−η1,3+c123+δ1 =

17

Table 2: Unit Shipping Costs and Demands

Location 1 Location 2

Size 5 6 3 5

Demand 20 20 10 30

Facility 1 2 4 2 5

Facility 2 3 3 1.5 4

Table 3: Optimal Dual Prices (ηik)

Remnant Size Facility

k i = 1 i =2

13 20 -

11 19 19

10 17 -

8 11.5 15.5

7 8.6 -

6 8.55 9.55

5 8.50 9.45

4 3.05 -

3 3 3.5

2 0.1 0.2

1 .05 0.1

18

8.55−3+2+0.05 = 7.60 and facility 2 has: η2,6−η2,3+c223+δ2 = 9.55−3.5+1.5+0.05 = 7.60.We would thus use the 13-unit piece at Facility 1 to fulfill this demand. On the other hand

if we only had 6-unit remnants at both locations, we would be indifferent between the two.

While this is a small example, it nevertheless illustrates the kinds of policies that we can

derive from the duality results.

2.3.3 Degeneracy, Multiple Optima, and Perturbation

In Section 2.2 at Chapter 2, we noted with an illustrative example that the primal problem

will generally have multiple optima. When attempting to adapt the static optimum to a

dynamic policy, an issue that needs to be addressed is that the problem tends to be degenerate

at optimality, with several of the yim,n variables being basic at zero values. Each of the optimal

bases leads to a different dual vector. Note that this should be intuitively obvious based

on the fact that most dual variables have objective coefficient of zero. Unfortunately, not

all of these dual vectors yield values for the optimal dual prices that are desirable from the

perspective of price-directed policies. There is no guarantee that an arbitrary dual vector will

satisfy properties that are intuitively appealing with the interpretation of the dual price for

a remnant as its inherent value. Adelman and Nemhauser [2] present several such properties

in their work on the single location problem with no cutting costs and no scrap values. We

present monotonicity and superadditivity as follows:

Definition 1. A value function η is monotonic if for any facility i ∈ I and any sizesm,m− n ∈ Ki, ηi,m ≥ ηi,m−n.

Definition 2. A value function η is superadditive if for any facility i ∈ I and any sizesm,m− n, n ∈ Ki, ηi,m ≥ ηi,m−n + ηi,n.

We now generalize the above concepts as follows:

Definition 3. For a given cost δ ∈

Consider any piece that has been cut into two smaller pieces. If a value function is δ-

monotonic, then the inherent value of this piece plus the unit cutting cost is at least as much

as the inherent value of any of the smaller pieces. If a value function is δ-superadditive, then

the inherent value of the piece plus the unit cutting cost is at least as much as the sum of

the inherent values of the two smaller pieces.

When δ = 0 for all facilities, the notions of δ-monotonicity and δ-superadditivity reduce

to classical notions of monotonicity and superadditivity. The interested reader is referred to

Adelman and Nemhauser [2] for a further discussion of these types of properties when δ = 0

for a single facility problem. It suffices to say that such properties should be satisfied by the

value function if we plan to use them for a dynamic policy, but picking an arbitrary optimal

dual vector for our problem does not necessarily guarantee this.

What we need, then, is a consistent way of picking the “correct” values for the optimal

dual variables for use in a dynamic environment where one must make decisions over time

based on these inherent values. To address this issue we adapt the approach followed in

Adelman and Nemhauser [2] of developing a so-called “²-perturbation” that creates small

exogenous supplies and demands to ensure positive flows in optimal arcs. Our perturbation

differs from the one in Adelman and Nemhauser [2] in several respects. First, with multiple

facilities we cannot simply create exogenous demand to ensure positive flows, since this

demand may be completely satisfied by a single facility and still leave the flows in all the

others degenerate. Rather, one needs to account for each facility individually and ensure

that there is (at least some minimal amount of) production of all sizes at each facility so

that we avoid yim,n = 0. Second, unlike in Adelman and Nemhauser [2] we do not assume

that the relative proportions of raw sizes at each facility is prespecified. Rather, these are

decision variables in our problem. Finally, we do not necessarily create perturbations for

every size.

To introduce the perturbation we first extend the set D, which was defined earlier as

the union of all sizes demanded across all locations. We now define a set ∆ by adding to D

every other size that could be generated at a facility i, i.e.

∆ = K ∪D. (2.15)

20

We partition ∆ as follows,

Definition 5. Define

a) ∆0 = {m|m ∈ ∆,m < min{k|k ∈ D}},b) ∆1 = {m|m ∈ ∆,m ≥ min{k|k ∈ D}}.

Set ∆0 indexes scrap sizes while ∆1 indexes all sizes in the extended demand set ∆ that

are at least as large as the minimum demand size in the original problem.

Next suppose ² is some small positive constant.

Definition 6. The parameters of an ²-perturbation are defined as αim ∈

For the size k ∈ ∆\D, there is no original demand and thus, x∗ijk = 0. Therefore, weperturb constraint (2.4) as follows:

∑

(m,n)∈A′i|(m−n)=kyim,n = βik², for all i ∈ I and k ∈ ∆\D. (2.19)

Finally, we drop constraint (2.3) in the perturbed model.

We can rewrite the primal perturbed problem as follows:

Minimize∑

i∈I∑

h∈Si aihrih +∑

i∈I∑

{(m,n)∈Ai|n>0} δiyim,n +

∑i∈I

∑j∈J

∑k∈Dj cijkx

∗ijk

−∑i∈I∑

u∈Ki uσisiu

subject to (2.16), (2.17), (2.18), (2.19), and y, r, s ≥ 0.

The dual of the perturbed problem is:

Maximize∑

i∈I∑

k∈∆ πik(∑

j∈J x∗ijk + βjk²)−

∑i∈I

∑m∈∆ ηi,mαim²

Subject to (2.8), (2.9), (2.10), (2.11), and η, π unrestricted.

Even if the size k ∈ Ki is not generated at location i for shipment to some demandpoint (i.e., xijk is zero for all j), the constraints in the above perturbed model still force

production of a small amount (βikε) of this size. For sizes that are generated for shipment,

the production is simply increased by a small amount.

Before getting into details about what it accomplishes, we examine the perturbation

further and contrast the optimal flows in the original problem with those in the perturbed

problem. Note that in the perturbed problem, for each m that belongs to ∆:

• There is an extra inflow of αim² units into node m.• There is an additional demand of βim² units for size m, where βim < αim if m ∈ ∆0 and

βim = αim if m ∈ ∆1.

We first show that in the optimal solution to the perturbed problem there is always a

positive flow from all nodes with scrap sizes to node 0.

22

Proposition 1. For every optimal solution ∗y to the perturbed problem, ∀i ∈ I, m ∈ ∆0:∗ym,0 = βim².

In order to prove Proposition 1, we first present the following definition and lemma.

Definition 7. Define P im,n as the set of directed paths p that start at node m and end at

node n in network G′i = (K′i, A

′i) at facility i ∈ I. Define ∗qip = min(m,n)∈p ∗yim,n, that is, the

optimal path flow along p, and ∗Qim,n =∑

p∈P im,n∗qip.

Lemma 1. Under an ²-perturbation, ∀i ∈ I, m ∈ Ki, ∗Qim,0 ≥ βim².

Proof. Suppose at the optimum ∗Qim,0 < βim², for some i ∈ I,m ∈ Ki. First, note that thetotal flow along all arcs (m1,m2) ∈ A′i such that m1 −m2 = m must be at least as much asβim², so that

∑

(m1,m2)∈A′i|m1−m2=m

∗yim1,m2 ≥ βim² > ∗Qim,0.

Isolating the flow along arc (m, 0), this implies that

∗yim,0 +∑

(m1,m2)∈A′i|m1−m2=m,m1 6=m

∗yim1,m2 >∗Qim,0.

But ∗Qim,0 ≥ ∗yim,0 by definition. Therefore,

∑

(m1,m2)∈A′i|m1−m2=m,m1 6=m

∗yim1,m2 > 0.

That is, there exists at least one m′ > m such that ∗yim′,m′−m > 0.

Now, since the inflow into node m is at least αim², it follows by flow balance that the out-

flow from node m,∑

(n|n∈K′i,0≤n

(b) Increase ∗yim,0 by ν with no added cutting cost. This increases production of size m

by ν units.

(c) Compensate for the extra outflow from node m by decreasing scrap ∗sim > 0 by ν

units.

(d) Compensate for the decreased outflow from node m′ by increasing scrap ∗sim′ by ν.

Case 2): Suppose ∗sim = 0, so that∑

(n|n∈K′i,0 0. Consider the following alternative

scheme:

(1) Reduce ∗yim′,m′−m by some small amount ν, thus saving νδi units in cutting costs.

This reduces production of size m by ν units.

(2) Increase ∗yim,0 by ν with no added cutting cost. This increases production of size m

by ν units.

(3) Compensate for the extra outflow from node m by decreasing ∗Qim,n by ν units and

save at least νδi units of cutting costs.

(4) Compensate for the decreased outflow from node m′ by increasing ∗Qim′,m′−(m−n) by

ν units, and add cutting costs exactly equal to those saved in Step (3).

Comparing either new cutting scheme with the original one, we find: 1) Total production

of sizes are the same in each scheme; 2) Either of the new cutting schemes will save costs

from the original cutting scheme. This contradicts the assumption that the original cutting

scheme is optimal. Thus, ∗Qim,0 ≥ βim².

We are now ready to prove Proposition 1.

Proof of Proposition 1: Given m ∈ ∆0, ∗yim,0 ≤ βim² since m is a scrap size. Suppose∗yim,0 < βim². Since

∗Qim,0 ≥ βim² by Lemma 1, it follows that there is at least one pathp ∈ P im,0 with intermediate nodes between node m and node 0 along which there are positiveflows. Consider one such path with a flow of ∗qip units. Assume the path consists of b arcs

that produce ∗qip pieces of sizes k1, k2, ..., kb where∑b

j=1 kj = m. Since all of these sizes

are scrap sizes and the total demands for these are βi(m−k1)² , ..., and βikb², it follows that

∗qip ≤ min1≤j≤b(βikj²). Consider ν such that 0 < ν ≤ ∗qip, and the following alternativescheme:

24

(a) Reduce the flow along the path by ν units and increase ∗sim by ν units, thus save

(b− 1)νδi units in cutting costs.(b) Increase ∗yim−kj ,0 for j = 1, 2, ..., b by ν units and add no cutting cost (note that this

is always possible since αi(m−kj)² > βi(m−kj)² ≥ ν for each node in the path).This will save (b−1)νδi units of cutting costs and satisfy all demands, which contradicts

the assumption of optimality. Therefore, ∗yim,0 = βim² and the additional demand of βim²

units for size m are satisfied at no additional cost. The remainder of (αim²− βim²) units isscrapped at node m. 2

Now suppose optimal flows in the original and the perturbed problem are given by

the vectors ŷ and ∗y respectively, while the corresponding optimal values of the objectives

are given by ẑ and ∗z. Assuming that cutting costs are strictly positive and that we are

not charged for the additional inflows of αim², it is clear that the optimal solution to the

original problem with the following modifications will be an optimal solution to the perturbed

problem:

• For m ∈ ∆1, the extra inflow of αim² units is routed along arc (m, 0), i.e., ∗ym,0 =ŷm,0 + αim² = ŷm,0 + βim². Thus it satisfies the additional demand of βim² units for size

m at no additional cost since there is no cutting.

• For m ∈ ∆0, a portion βim² of the extra inflow of αim² units is routed along arc (m, 0),i.e., ∗ym,0 = ŷm,0 + βim² = βim². Thus it satisfies the additional demand of βim² units for

size m at no additional cost. The remainder of (αim² − βim²) units is scrapped at nodem since there is no original demand for size m.

Thus we do not pay any additional cutting costs in the perturbed problem and ∗z =

ẑ −∑i∈I,m∈∆0(αim²− βim²)σi. We can see that, as ε → 0 this ε-perturbation reduces to theoriginal problem.

To clarify and illustrate the perturbation and its optimal solution, consider facility i with

Si = {11} and D = {3, 5} shown in Figure 2. Suppose that λ3 = 20 and λ5 = 10 and theoptimum solution has ŷ11,6 = ŷ6,3 = ŷ3,0 = 10 as shown in Figure 4.

For the ²-perturbation we have ∆ = {1, 2, 3, 5, 6, 8, 11} with ∆0 = {1, 2} and ∆1 ={3, 5, 6, 8, 11}, and we define αim = βim for m ∈ ∆1 and αim > βim for m ∈ ∆0. The

25

Figure 4: Optimal Cutting Scheme for Original Problem

Figure 5: Optimal Cutting Scheme for Perturbed Problem

node set for the perturbed problem is given by K ′i = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, and theoptimal flow for the perturbed problem described above is given by Figure 5.

While the above flows describe the optimum for the perturbed problem, they are not

necessarily unique. To illustrate this fact, consider the flow pattern depicted in Figure 6,

which differs from Figure 5 in the following flows only:

• ∗y8,0, which is now equal to zero,

26

Figure 6: Alternative Optimum for Perturbed Problem

• ∗y11,6, ∗y6,3, each of which is decreased by an amount βi8², and• ∗y11,3, ∗y8,3, ∗y3,0, each of which is increased by an amount βi8².

Note that flow balance is maintained and the net flow for each different size is identical

in both cases so that all demand is satisfied. We save 2(βi8²)δi corresponding to reduced

flows in arcs ∗y11,6 and ∗y6,3 that now need not be cut, but pay an additional 2(βi8²)δi due

to the increased flows along ∗y11,3 and ∗y8,3 and the corresponding extra cutting cost. Thus,

there is no overall change in costs and thus this also represents an optimal solution.

Rather than solving the perturbed problem directly, we will find it is convenient to

construct an optimal solution from the optimal solution to the unperturbed problem.

We also present an algorithm for constructing alternative optima when these exist. Al-

gorithm 1 constructs an optimal solution to the perturbed problem:

Algorithm 1 Obtaining an Optimal Solution to the Perturbed Primal Problem

1: Solve the unperturbed problem and obtain the optimal solution ŷ, with optimal value ẑ.

2: For m ∈ ∆1, let ∗ym,n = ŷm,n for n > 0 and ∗ym,0 = ŷm,0 + αim² = ŷm,0 + βim².3: For m ∈ ∆0, let ∗ym,0 = βim² and scrap the remainder of (αim²− βim²) units at node m.

Proposition 2. The solution obtained by Algorithm 1 is optimal for the perturbed problem.

Proof. For m ∈ ∆0, the conclusion follows from Step 2 of Algorithm 1 and Proposition 1.

27

For m ∈ ∆1, we create exactly as much of supply (αim²) as the new demand (βim²). Byassigning all of the extra supply to the arc (m, 0), we create exactly enough of size m to

satisfy all additional demands at no extra cost, from which the conclusion follows.

Corollary 1. Let ẑ be the optimal value for the unperturbed problem. Then the optimal

value ∗z for the perturbed problem is: ∗z = ẑ −∑i∈I,m∈∆0(αim²− βim²)σi.

Suppose that in the optimal solution created by Algorithm 1, ∃m,m′ ∈ ∆1, k1, k2, ..., kp ∈D with k1 + k2 + ... + kp = m,m

′ > m such that ∗yim′,m′−k1 > 0,∗yim′−k1,m′−k1−k2 > 0, ...,

∗yim′−Pa=p−1a=1 ka,m′−m

> 0. Then we can find alternative optimal solutions for the perturbed

problem by Algorithm 2:

Algorithm 2 Obtaining Alternative Optimal Solutions to the Perturbed Primal Problem

1: Apply Algorithm 1 and obtain an optimal solution ∗y.

2: Reduce the flow along the path (m′,m′ − k1), (m′ − k1,m′ − k1 − k2), ..., (m′ −∑a=p−1

a=1 ka,m′ −m) by some small amount ν ≤ βim².

3: Compensate for the reduced production of sizes k1, k2, ..., kp by increasing the flow along

the path (m,m− k1), (m− k1, m− k1 − k2), ..., (m−∑a=p−1

a=1 ka, 0) by ν.

4: Compensate for the increased flow out of node m by reducing the flow along arc (m, 0)

by ν, thus producing ν fewer pieces of size m.

5: Compensate for the decreased flow out of node m′ by increasing the flow along the arc

(m′,m′ −m) by ν, thus producing ν additional pieces of size m.

We now show that the solution obtained by Algorithm 2 is also optimal.

Proposition 3. Suppose that in the optimal solution created by Algorithm 1, ∃m,m′ ∈∆1(m

′ > m), k1, k2, ..., kp ∈ D with k1 + k2 + ... + kp = m such that ∗yim′,m′−k1 > 0,∗yim′−k1,m′−k1−k2 > 0, ...,

∗yim′−Pa=p−1a=1 ka,m′−m

> 0. Then by applying Algorithm 2 for ∀m ∈∆1, we can create alternative optimal solutions to the perturbed problems at each facility

i, i ∈ I with the following properties∀m ∈ ∆1,

• ∗yim′,m′−m > 0;• ∗yim,m−k1 > 0, ∗yim−k1,m−k1−k2 > 0, ..., ∗yim−Pa=p−1a=1 ka,0 > 0.

28

Proof. In Step 2 of Algorithm 2, we save νp units in cutting costs.

In Step 3 of Algorithm 2, we pay ν(p− 1) units in cutting costs.In Step 4 of Algorithm 2, no cutting costs are added.

In Step 5 of Algorithm 2, we pay ν units in cutting costs.

Thus Algorithm 2 incurs exactly the same cutting costs as Algorithm 1 while producing

the same amount of products. Consequently, the solution created by Algorithm 2 is optimal.

Note that Figure 5 illustrates an optimal solution by Algorithm 1 while Figure 6 illus-

trates an alternative optimal solution by Algorithm 2 with p = 2, m = 8,m′ = 11 and

k1 = 3, k2 = 5. The following proposition characterizes any solution to the perturbed prob-

lem.

Proposition 4. ∀i ∈ I, m ∈ ∆1, any solution to the perturbed problem must have thefollowing characteristics,

a) ∗yim,0 ≥ βim² > 0, orb) 0 ≤ ∗yim,0 < βim², ∃m′ ∈ Ki,m′ > m, and k1, k2, ..., kq ∈ Ki with

∑qg=1 kg = m such

that

• ∗yim′,m′−m > 0;• ∗yim,m−k1 > 0, ∗yim−k1,m−k1−k2 > 0, ..., ∗yi(m−Pq−1g=1 kg),0 > 0.

Proof. ∀m ∈ ∆1: From Lemma 1, ∗Qim,0 =∑

p∈P im,0∗qip =

∗yim,0 +∑

p∈(P im,0\(m,0))∗qip ≥ βim².

We consider the two cases: ∗yim,0 ≥ βim² and ∗yim,0 < βim².a) If ∗yim,0 ≥ βim² > 0, the result follows immediately.b) Suppose ∗yim,0 < βim² so that

∑p∈(P im,0\(m,0))

∗qip > 0. Then there must exist at

least one path with flow amount ν such that 0 < ν ≤ βim² − ∗yim,0 and k1, k2, ..., kp withk1 + k2 + ... + kp = m such that

∗yim,m−k1 > 0,∗yim−k1,m−k1−k2 > 0, ...,

∗yi(m−Pp−1g=1 kg),0

> 0.

Furthermore, since ∗yim,0 < βim², it follows that ∃m′ > m such that ∗yim′,m′−m > 0 tosatisfy the additional demand for size m.

To summarize the need for the preceding description, we note that because of dual

degeneracy we may obtain inherent values that may not satisfy the properties of monotonicity

29

and superadditivity, which makes a price-directed policy inapplicable. We therefore develop

and solve a perturbed model. In the next section, we show that the inherent values from

the perturbed model will satisfy these describe properties; Section 2.4 will be used for this

purpose.

2.4 PROPERTIES OF THE ²-PERTURBATION

We now prove some important properties of the static problem that will hold under an

²-perturbation, and thus make our policy applicable in a dynamic environment.

Proposition 5. The perturbed problem is always feasible.

Proof. The conclusion follows from the fact that if right hand sides of constraints (2.2) and

(2.4) are positive, the constraints can always be satisfied by adjusting the raw material

consumption because there are no upper bounds for these: there are no capacity limits at

facilities.

Denote an optimal primal solution to the perturbed problem by (∗y, ∗r, ∗s) and the cor-

responding optimal dual multipliers by (∗π, ∗µ, ∗η). Recall that we also assume that the

cutting cost δi is strictly positive.

The following proposition proves that permutability holds, i.e. satisfying a demand for

size k1 by cutting node m and then cutting the remnant to satisfy a demand for size k2,

is equivalent to satisfying the demand for size k2 first and then k1. Interested readers may

refer to the paper by Adelman and Nemhauser [2] for a discussion of permutability property

when cutting costs are zero.

Proposition 6. For any i ∈ I, k1, k2 ∈ ∆, and m,m − k1, m − k1 − k2 ∈ K ′i, suppose∗yim,m−k1 > 0 and

∗yim−k1,m−k1−k2 > 0. Then

a) ∗πik1 =∗ηi,m − ∗ηi,m−k1 + δi, ∗πik2 = ∗ηi,m−k1 − ∗ηi,m−k1−k2 + δi.

b) ∗πik2 =∗ηi,m − ∗ηi,m−k2 + δi, ∗πik1 = ∗ηi,m−k2 − ∗ηi,m−k2−k1 + δi.

30

Proof. The conclusion of part a) directly follows from complementary slackness corresponding

to the assumption that there exist ∗yim,m−k1 > 0 and∗yim−k1,m−k1−k2 > 0.

To show the conclusion of part b), by complementary slackness corresponding to (2.8),

it follows that

∗πik1 =∗ηi,m − ∗ηi,m−k1 + δi, (2.20)

and

∗πik2 =∗ηi,m−k1 − ∗ηi,m−k1−k2 + δi. (2.21)

Constraint (2.8) for arcs (m, m− k2) and (m− k2,m− k1 − k2) implies that

∗ηi,m − ∗ηi,m−k2 + δi ≥ ∗πik2 , (2.22)

and

∗ηi,m−k2 − ∗ηi,m−k1−k2 + δi ≥ ∗πik1 . (2.23)

(2.21) and (2.22) imply that

∗ηi,m−k2 ≤ ∗ηi,m − ∗ηi,m−k1 + ∗ηi,m−k1−k2 , (2.24)

while (2.20) and (2.23) imply that

∗ηi,m−k2 ≥ ∗ηi,m − ∗ηi,m−k1 + ∗ηi,m−k1−k2 . (2.25)

Then, from (2.24) and (2.25) it follows that

∗ηi,m−k2 =∗ηi,m − ∗ηi,m−k1 + ∗ηi,m−k1−k2 . (2.26)

Equation (2.26) can then be written in the following two ways using (2.21) and (2.20)

respectively:

∗ηi,m − ∗ηi,m−k2 + δi = ∗ηi,m−k1 − ∗ηi,m−k1−k2 + δi = ∗πik2 ,

and

∗ηi,m−k2 − ∗ηi,m−k1−k2 + δi = ∗ηi,m − ∗ηi,m−k1 + δi = ∗πik1 ,

from which the conclusion follows.

31

The following proposition assigns an appropriate inherent value to scrap sizes.

Proposition 7. Under an ²-perturbation, ∀i ∈ I, m ∈ ∆0, ∗ηi,m = mσi.

Proof. The total inflow into node m is∑

(n,m)∈A′i∗yin,m + αim² and it is equal to the outflow

from node m,∑

(m,n)∈A′i∗yim,n +

∗sim. Consider ∗yim,n, ∀(m,n) ∈ A′i. From Proposition1, ∗yim,0 = βim². Further,

∗yim,n = 0 for all other n since m − n < m has no originaldemand. Thus,

∑(m,n)∈A′i

∗yim,n +∗sim = βim² + ∗sim =

∑(n,m)∈A′i

∗yin,m + αim² ≥ αim².But by Definition 6, βim² < αim². It follows that

∗sim > 0. The result then follows from

complementary slackness corresponding to (2.11).

Proposition 9, which we introduce next, shows that δ-superadditivity will always hold

with the ²-perturbation. To prove this proposition, we first introduce the following crucial

proposition, which will also be used subsequently to prove δ-monotonicity.

Proposition 8. Under an ²-perturbation, ∀i ∈ I, m ∈ Ki, ∗πim = ∗ηi,m.

Proof. Consider an optimal primal solution ∗y to the perturbed problem.

For m ∈ ∆0, ∗yim,0 > 0 by Proposition 1. Complementary slackness corresponding to(2.9) implies that ∗πim = ∗ηi,m − ∗ηi,0. By Proposition 7, ∗ηi,0 = 0, and thus, ∗πim = ∗ηi,m.

For node m ∈ ∆1, there are two possibilities:Case 1): ∗yim,0 > 0. The proof is identical to the proof for m ∈ ∆0.Case 2): ∗yim,0 = 0. From Proposition 4, ∃m′ > m such that ∗yim′,m′−m > 0 to satisfy the

additional demand for size m. Let l′ denote a raw size, from which there is a flow into node

m′.

Proposition 4 also states that there exists a positive flow along the path p = {(m,m−k1),(m − k1,m − k1 − k2), ..., (m −

∑q−1g=1 kg, 0)}, from which we obtain products with sizes

k1, k2, ..., kq. Note that sizes k1, k2, ..., kq must correspond to sizes that have original demand.

From Corollary 1, it follows that the optimal solution to the perturbed problem must

not use additional original raw material, that is, the optimal solution to the perturbed

problem uses the same amount of all original raw material as the one to the unperturbed

problem. Thus, flow along the path p must free up exactly the same amount (say ν units)

of the same size of raw material (i.e. raw size l′) as the extra requirement of the flow

32

along the arc (m′,m′ −m). To ensure this, there must exist m′′ such that ∗yim′′,m′′−k1 > 0,∗yim′′−k1,m′′−k1−k2 > 0, ...,

∗yi(m′′−Pq−1g=1 kg),m′′−m

> 0 in the path from l′ to m. Note that m′′

may or may not be identical to node m′.

By complementary slackness for (2.8), the following q equalities follow

∗ηi,m′′ − ∗ηi,m′′−k1 + δi = ∗πik1 .∗ηi,m′′−k1 − ∗ηi,m′′−k1−k2 + δi = ∗πik2 .

...

∗ηi,(m′′−Pq−1g=1 kg) −∗ηi,m′′−m + δi = ∗πikq . (2.27)

Adding all equations of the form (2.27), we obtain

∗ηi,m′′ − ∗ηi,m′′−m + qδi = ∗πik1 + ... + ∗πikq . (2.28)

Also, ∗yim,m−k1 > 0, ...,∗yi

(m−Pq−2g=1 kg),(m−Pq−1

g=1 kg)> 0 and complementary slackness corre-

sponding to (2.8) imply that

∗ηi,m − ∗ηi,m−k1 + δi = ∗πik1 ,

...

∗ηi,(m−Pq−2g=1 kg) −∗ηi,(m−Pq−1g=1 kg) + δi =

∗πikq−1 , (2.29)

while ∗yi(m−Pq−1g=1 kg),0

> 0 and complementary slackness corresponding to (2.9) imply that

∗ηi,(m−Pq−1g=1 kg) −∗ηi,0 = ∗πikq . (2.30)

Adding all equations of the form (2.29) to equation (2.30) and recalling that ∗ηi,0 = 0

from Proposition 7, it follows that

∗ηi,m + (q − 1)δi = ∗πik1 + ... + ∗πikq . (2.31)

Finally, ∗yim′,m′−m > 0 and complementary slackness corresponding to (2.8) imply that

∗ηi,m′ − ∗ηi,m′−m + δi = ∗πim. (2.32)

33

By Proposition 6, we obtain

∗ηi,m′′ − ∗ηi,m′′−m + δi =∗ ηi,m′ − ∗ηi,m′−m + δi = ∗πim. (2.33)

Comparing equations (2.28) and (2.33), we conclude

∗πim + (q − 1)δi = ∗πik1 + ... + ∗πikq . (2.34)

From equations (2.31) and (2.34), it then follows that

∗πim = ∗ηi,m, (2.35)

thus concluding the proof.

This leads to the following proposition:

Proposition 9. Let (∗η, ∗π) be optimal dual multipliers under an ²-perturbation. Then the

value function ∗η is δ-superadditive.

Proof. Consider any nodes l, m, l+m ∈ Ki at an arbitrary facility i ∈ I. From dual constraint(2.8), ∗ηi,l+m + δi − ∗ηi,m ≥ ∗πil, and by Proposition 8, ∗πil = ∗ηi,l. Thus, ∗ηi,l+m + δi ≥∗ηi,l + ∗ηi,m.

Next, we address δ-monotonicity and show that it follows as a corollary from the following

proposition, which provides a general relationship between the inherent values of different

sizes at a facility.

Proposition 10. Let (∗η, ∗π) be optimal dual multipliers under an ²-perturbation. Then

∗ηi,m + δi − σi(m− n) ≥ ∗ηi,n, ∀i ∈ I, m, n ∈ Ki such that m− n ∈ Ki.

Proof. ∗ηi,m + δi − (m− n)σi ≥ ∗ηi,m + δi − ∗ηi,m−n ≥ ∗πi,n where the first inequality followsfrom dual constraint (2.11), and the second follows from constraint (2.8). By Proposition 8,

∗πi,n = ∗ηi,n, from which the proposition follows.

Proposition 10 indicates that the inherent value of a remnant depends on both cutting

costs and scrap value. Two corollaries follow from this: Corollary 2 shows that as long as

34

cutting costs are positive, δ-monotonicity follows directly, while Corollary 3 shows that if

the unit scrap value at each facility is higher than the unit cutting cost at the facility, then

the inherent value is strictly decreasing with size.

Corollary 2. Let (∗η, ∗π) be optimal dual multipliers under an ²-perturbation. Then the value

function ∗η is strictly δ-monotonic, that is, ∗ηi,m + δi > ∗ηi,n, ∀i ∈ I, m, n(m > n) ∈ Ki.

Proof. Follows directly from Proposition 10 and the fact that (m− n)σi > 0.

Corollary 3. Under an ²-perturbation, ∀i ∈ I, if σi ≥ δi, then ∗ηi,m ≥ ∗ηi,n, ∀m > n, m, n ∈Ki.

Proof. If σi ≥ δi, then since m − n ≥ 1, δi − σi(m − n) ≤ 0. Then, Proposition 10 impliesthat ∗ηi,m ≥ ∗ηi,n.

Conversely, if the cutting cost is high, it is possible that (m−n)σi− δi ≤ ηi,m− ηi,n < 0,so that the inherent values could actually increase as the size decreases.

The next proposition that we introduce provides an upper bound on the inherent value

of a specific size at a specific facility. Suppose that for any given raw stock of size h and

remnant of size l, we denote dh,l as the maximum number of pieces of the remnant that can

be cut from one piece of raw stock, d′h,l as the number of cuts made to obtain dh,l pieces,

and bh,l as the scrap size remaining after cutting. That is, dh,l = bh/lc, bh,l = h− ldh,l, andd′h,l = dh,l if bh,l > 0, d

′h,l = dh,l−1 otherwise. Then we obtain a bound on the inherent value

as follows:

Proposition 11. Under an ²-perturbation, ∀i ∈ I, l ∈ Ki, ∗ηi,l ≤ min(h∈Si,h≥l){aih + d′h,lδi −bh,lσi}/dh,l.

Proof. Consider an arc from node h (h ≥ l) to node m = h− l, where h ≥ l. By constraint(2.8),

∗ηi,h + δi ≥ ∗ηi,h−l + ∗πil.

Adding (d′h,l − 1)δi to both sides yields

∗ηi,h + d′h,lδi ≥ ∗ηi,h−l + ∗πil + (d′h,l − 1)δi.

35

Since ∗ηi,h ≤ aih from (2.10), and ∗πil = ∗ηi,l from Proposition 8, it follow that

aih + d′h,lδi ≥ ∗ηi,h−l + ∗ηi,l + (d′h,l − 1)δi.

Now consider the arc from node h− l to node h− 2l. Adding ∗ηi,l + (d′h,l − 2)δi to bothsides of constraint (2.8) for this arc and using Proposition 8 we obtain

∗ηi,h−l + ∗ηi,l + (d′h,l − 1)δi ≥ ∗ηi,h−2l + 2∗ηi,l + (d′h,l − 2)δi.

If we continue the same procedure through nodes h− 2l, h− 3l, ... we eventually reachthe arc from node l + bh,l to node bh,l, where size bh,l is either zero or corresponds to a scrap

size. The constraint for this arc (constraint (2.8) if bh,l > 0 or constraint (2.9) if bh,l = 0)

yields

∗ηi,h−(dh,l−1)l + (dh,l − 1)∗ηi,l + (d′h,l − (dh,l − 1))δi ≥ ∗ηi,bh,l + dh,l∗ηi,l.

The above sequence of relationships implies that

aih + d′h,lδi ≥ ∗ηi,bh,l + dh,l∗ηi,l.

By Proposition 7, ∗ηi,bh,l = bh,lσi, it follows that

aih + d′h,lδi ≥ bh,lσi + dh,l∗ηi,l,

from which the proposition follows.

Based upon computational tests, it appears that on average, these bounds are between

approximately 7.10% (for small problems) and 11.32% (for large problems) higher than the

true values. Details may be found in Section 2.5.

Finally, the following proposition provides a bound on the difference in inherent values

between the same sizes but at different facilities:

Proposition 12. If the scrap value is the same for each facility, then under ²-perturbation,

|∗ηi,l − ∗ηi′,l| ≤ minh∈Si,h′∈Si′{aih − hσ + δi, ai′h′ − h′σ + δi′}.

36

Proof. Consider any raw size h, and any remnant l ((h, l) ∈ Ai) at facility i (i ∈ I). Fromdual constraint (2.8), it follows that

∗ηi,l ≤ aih − ∗ηi,h−l + δi. (2.36)

constraint (2.11) and the above inequality yield

∗ηi,l ≤ aih − (h− l)σ + δi. (2.37)

If we consider a similar constraint for size l at facility i′, it follows that

∗ηi′,l ≤ ai′h′ − (h′ − l)σ + δi′ . (2.38)

From dual feasibility, we obtain

lσ ≤ ∗ηi′,l, (2.39)

and

lσ ≤ ∗ηi,l. (2.40)

From (2.37) and (2.39), we obtain

∗ηi,l − ∗ηi′,l ≤ aih − hσ + δi. (2.41)

Similarly, (2.38) and (2.40) imply that

∗ηi′,l − ∗ηi,l ≤ ai′h′ − h′σ + δi′ . (2.42)

From (2.41) and (2.42), the result then follows.

To check the performance of our policy based on the inherent values, we develop a

simulation model to compare our policy with two other policies. The results are described

in the following chapter.

37

2.5 SIMULATION AND INVENTORY CONTROL

We designed and ran a simulation model to compare our policy with two other heuristic

policies to test how our adaptation of the optimal static policy performs in a dynamic and

stochastic demand environment. We assume this supply chain is operated over a long period

of time. In each period (day), demand for various sizes arise at various demand location.

These demands arrive randomly according to independent Poisson processes with rate λjk

for size k at location j. There are no capacity constraints on the facility and all demand is

to be fulfilled.

Our goal was to test our price-directed policy against two other plausible heuristic poli-

cies. Each of the three policies is described briefly below:

A. Smallest Fit: This is a greedy approach where we search all facilities for the smallest

remnant or raw piece that can accommodate the demanded size. Ties are broken by picking

the option that minimizes the sum of raw materials costs (if any), shipping costs, and cutting

costs, less the value of any scrap generated.

B. Multi-criteria Heuristic: This policy follows a hierarchy of choices. First, if a

remnant of the exact size demanded exists, it is used. If not, we look for the smallest

feasible remnant that can not be used after cutting the current demand size. If no such

piece exists, we cut from the largest feasible remnant in stock (thus leaving another smaller

useful remnant behind). Finally, if there are no feasible remnants, we cut from the largest

raw stock size. In all instances, ties are broken using the same criterion as in Policy A.

C. Price-Directed Policy: This is our policy that uses dual values from the perturbed

version of the static LP formulation. We look across all facilities and pick the facility i and

size m according to the criterion described in Theorem 1, where ties are broken arbitrarily.

An important consideration with a price-directed policy is that inventories must be suitably

controlled; without constraints on raw materials it is possible that inventories of certain rem-

nant sizes can grow unchecked. Adelman and Nemhauser [2] get around this by empirically

setting a suitable (and identical) limit on the inventory of each size.

We follow a different approach. For each day, we track the number of pieces of all sizes

created, as well as the number of pieces of each raw size used during that day at every site.

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These numbers are then compared to the optimal yim,n and rih values from the solution to

the static LP in order to set limits on their inventory as follows: suppose the demand to be

met is for size k, so that cutting a piece from the selected size m would leave a remnant of

size n = m−k. If the size m corresponds to a remnant we check to see if the total number ofpieces of size n created after this operation exceeds the optimal daily rate from the static LP

for size n (i.e., exceeds the sum of the flow into or out of node n). If the size m corresponds

to a raw size we check to see if the usage of that raw size exceeds the optimal daily rate rim.

If either of these occurs we forego the option of using facility i and size m, and move on to

the next best one according to the inherent value criterion. The only time we violate these

inventory restrictions is if it is completely impossible to otherwise satisfy the demand. In

that case we just pick the location and raw size that yield the best value for the inherent

value criterion.

To compare these policies we tested three randomly generated sets of instances using

our simulation model: these consist of 10 small, 10 medium and 10 large-size instances

respectively. Table 4 provides some key characteristics of each instance set. For each of the

thirty instances, the actual demand for each size at each demand location over a 300-da